Trade-Offs in MIMO Radar With Colocated Antennas · Trade-Offs in MIMO Radar With Colocated Antennas ... waveform diversity pattern R(θ): Rx coherent processing pattern ... beneﬁts

Intro T/R Beamforming Tr Energy Focusing Tr Array Mapping Localization Jammers Suppresstion TB AF Ref Contr

Trade-Offs in MIMO Radar With ColocatedAntennas

Sergiy A. Vorobyov

SAM Workshop


Introduction

MIMO radar is an emerging concept that has recently at-tracted a lot of interest

MIMO radar: A radar system with multiple transmit wave-forms that is able to jointly process signals received at multi-ple receive antennas

A MIMO radar may be configured with widely separated orwith co-located antennas (we focus on the latter)

MIMO radar has many advantages such as extended arrayaperture, improved resolution, enhanced identifiability, etc.

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Introduction (Cont’d)

Phased-array enables coherent processing yielding an M2NSNR gainAs compared to phased-array, MIMO radar suffers fromseveral drawbacks such as

A factor of M loss in SNR gain [Daum and Huang, 2009]A factor of M loss in clear area [Abramivich and Frazer, 2008]

Phased-MIMO radar enables applying the concept of MIMOradar without having to sacrifice the coherent processinggain

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Phased-MIMO Radar

Let {!k (t)}Kk=1 (1 ! K ! M) be a set of baseband orthonor-

mal waveformsThe complex envelope of the signals transmitted by M trans-mit antennas

!(t)! "# $M!1

=

%MK

W"!(t)!"#$K!1

!(t) = [!1(t), . . . ,!K (t)]: K " 1 waveform vectorW = [w1, . . . , wK ]: M " K transmit weight matrixwk : M " 1 weight vector used to form the k th Tx beamTx beamforming can be optimized over {wk}K

k=1 to satisfycertain Tx beampattern requirementstotal Tx power constraintsand/or Tx power per antenna constraints

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Phased-MIMO Radar (Cont’d)

We consider Tx array partitioning

W !

&

'''''''(

w1,1 0 . . . 0... w2,1 0K#2

w1,M#K+1...

... wK ,1

0K#2 w2,M#K+1...

0 0K#2 wK ,M#K+1

)

*******+

Under fixed total transmit energy, the signal transmitted bythe k th subarray

sk (qT + t) =%

MK

w"k!k (t)

wk : (M-K+1)" 1 unit-norm weight vectorEsk = M/K : total Tx energy constraint (ET = M)

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Signal reflected from target

r(qT + t) !%

MK"t(q)

K,

k=1

e#!"k (#t )-

wHk ak (#t)

.!k (t)

"t(q): target reflection coefficient (changes from pulse topulse)$k (#t): propagation time-delay between 1st element of 1stsubarray and 1st element of k th subarraysThe N " 1 received data snapshot

x(qT + t) =

%MK"t(q)

K,

k=1

e#!"k (#t )-

wHk ak (#t)

.b(#t)!k (t)

+xi+n(qT + t)

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Using K matched filters

xk (q) !/

Tx(qT + t)!"

k (t)dt , k = 1, . . . ,K

Stacking {xk (q)}Kk=1 yields the KN " 1 virtual snapshot

y(q) =%

MK"t(q)u(#t) + yi+n(q) (1)

u(#) is the KN " 1 virtual steering vector

u(#) !

0

12wH

1 a1(#)e#!"1(#)

...wH

K aK (#)e#!"K (#)

3

45# b(#). (2)

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For K = 1, (1) and (2) simplify to the phased-array while thecase K = M corresponds to MIMO radar.

The benefits of MIMO radar can be achieved at a higherSNR gain.

Tx beamforming can be optimized over {wk}Kk=1 to satisfy

certain Tx beampattern requirements.Tx power constraints, e.g. per antenna

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Non-adaptive Tx/Rx Beamforming

We analyze the performance of the proposed formulations in thecontext of conventional Tx/Rx beamforming

Tx/Rx beamforming weights

wk =1$

M % K + 1ak (#t)

wd = u(#t)

Normalized Tx/Rx Beampattern

GK (#) !66wH

d u(#)662

66wHd u(#t)

662=

66u(#t)Hu(#)662

&u(#t)&2

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Phased-MIMO Tx/Rx Beampattern

For simplicity, consider ULAe!!"k (#) = a[k ](#)wH

1 a1(#) = . . . = wHK aK (#)

Rewriting the virtual steering vector

u(#) =

0

12wH

1 a1(#)e#!"1(#)

...wH

K aK (#)e#!"K (#)

3

45# b(#) = wHK aK (#)[a(#)# b(#)]

a(#): K " 1 vector (first K entries of a(#))aK (#): last M % K + 1 entries of a(#)

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Phased-MIMO Tx/Rx Beampattern (Cont’d)

Rewrite the beampattern expression as

GK (#) =

66aHK (#t)aK (#)

662

&aK (#t)&2! "# $

CK (#)

·66aH(#t)a(#)

662

&a(#t)&2! "# $

DK (#)

·66bH(#t)b(#)

662

&bK (#t)&2! "# $

R(#)

(3)

CK (#): Tx coherent processing patternDK (#): waveform diversity patternR(#): Rx coherent processing pattern (does not depend on K )

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Phased-MIMO Tx/Rx Beampattern (Cont’d)

By inspecting (3), we haveFor K = 1 (Phased-array radar)

C1(#) =

66aH(#t)a(#)662

&aH(#t)&2 , D1(#) = 1

Maximum Tx coherent gain, no diversity gainFor K = M (Traditional MIMO radar)

CM(#) = 1, DM(#) =

66aH(#t)a(#)662

&aH(#t)&2

No Tx coherent gain, Maximum waveform diversity gain

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Phased-MIMO Output SINR

The output SINR

SINR !MK %2

$t|wdu(#t)|2

wdRi+nwd=

MK %2

$t|uH(#t)u(#t)|2

Pi + %2n&u(#t)&2

Dominant noise power (Pi ' %2n)

SINRK (%2$t

%2n· M

K&u(#t)&2

= M(M % K + 1)N! "# $Gain

%2$t

%2n!"#$

SNR

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Phased-MIMO Output SINR (Cont’d)

Dominant noise power (Cont’d)Phased-array output SINR (K = 1)

SINRPH ( M2N%2$t

%2n

MIMO output SINR (K = M)

SINRMIMO ( MN%2$t

%2n

=1M

SINRPH

Phased-MIMO output SINR

SINRPH!MIMO ( M % K + 1M

" SINRPH (4)

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Output SINR (Cont’d)

Dominant interference power:Phased-array and MIMO radars have the same same side-lobe levels, hence

SINRMIMO ( SINRPH

Phased-MIMO radar is shown to have

SINRPH!MIMO ) SINRPH

via proving

SL

7sin K!

2

K sin !2·

sin (M!K+1)!2

(M % K + 1) sin !2

8! SL

7sin M!

2

M sin !2

8

! ! %d sin(#)&

SL{·} highest sidelobe level

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Phased-Array Tapering

Consider the special case when

!1(t) = . . . = !1(t) = !(t)

can be interpreted as phased-array with taperingApplying the same beamforming weights per each subar-ray yields the same beampattern and SINR gain as that ofphased-MIMO radarHowever, the waverform diversity and the correspondingbenefits are lost

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Example 1: Beampatterns

M = 10 sensors spaced half a wavelength apart. Samearray for Tx/Rx.Target located at #t = 10$

Two interferences located at %30$ and %10$.K = 5 subarrays9&k (t) = e!2% k

T t:K

k=1.

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Example 1: Transmit Beampattern

Transmit beampattern

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Example 1: Diversity Beampattern

Diversity beampattern

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Example 1: Overall Tx/Rx Beampattern

Overall Tx/Rx beampattern

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Example 2: Output SINR under weak interferencepower

INR = %30 dB

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Example 3: Output SINR under dominant interferencepower

INR = 30 dB

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DOA Estimation in MIMO Radar

DOA estimation for MIMO radar using ESPRIT [Duofang et.al.’08] and PARAFAC [Nion and Sidropoulos, ICASSP’09].In some applications, a priori information about the generalangular source sectors is available.Omni-directional transmission results in waste of energy inthe out-of-sector areas.Full-waveform diversity based MIMO radar suffers from SNRloss.

SolutionThe SNR per virtual antenna can be increased by

transmitting less waveforms of higher energy.focusing transmitted energy within certain spatial sectors.

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DOA Estimation in MIMO Radar

DOA estimation for MIMO radar using ESPRIT [Duofang et.al.’08] and PARAFAC [Nion and Sidropoulos, ICASSP’09].In some applications, a priori information about the generalangular source sectors is available.Omni-directional transmission results in waste of energy inthe out-of-sector areas.Full-waveform diversity based MIMO radar suffers from SNRloss.

SolutionThe SNR per virtual antenna can be increased by

transmitting less waveforms of higher energy.focusing transmitted energy within certain spatial sectors.

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Contents

A method for designing the transmit weight matrix, which isbased on maximizing the energy transmitted within the de-sired spatial sector and minimizing the energy disseminatedin the out-of-sector area, is developed.The proposed transmit energy focusing results in SNR im-provement at the receive array enabling better DOA estima-tion performance.An expression for the Cramer-Rao bound that shows itsdependence on the transmit weight matrix and the numberof transmitted waveforms is derived.

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MIMO Radar Signal Model

Transmitting M waveforms, l th target reflection is given by

rl(t , $) !%

EM

"l($)aT (#l)"(t).

Assuming L targets, the N "1 received vector is modeled as

x(t , $) =L,

l=1

rl(t , $)b(#l) + z(t , $).

Matched-filtering the received data to the transmitted wave-forms yields

y($) =

%EM

L,

l=1

"l($) a(#l)# b(#l)! "# $MN!1

+z($). (5)

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Proposed Formulations

The signal radiated from a beam towards the direction # canbe modeled as

s(t , #)=%

EK

cHk a(#)!k (t)

Forming K transmit beams, the N " 1 received vector ismodeled as

x(t , $)=%

EK

L,

l=1

'l($) ·;-

CHa(#l).T

"K (t)<

b(#l)+z(t , $).

Matched-filtering xbeam(t , $)to the k th transmitted waveforms

xk ($) !/

x(t , $)!"k (t)dt =

%EK

L,

l=1

'l($)-

cHk a(#l)

.b(#l) + zk ($)

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Proposed Formulations (Cont’d)

Stacking the individual vector components into one columnvector, the KN " 1 virtual data vector is obtained as

y($) ! [xT1 ($) · · · xT

K ($)]T

=

%EK

L,

l=1

'l($)-

CHa(#l).# b(#l) + zK ($). (6)

SNR improves due to|cH

k a|2: transmit beamforming gain.E/K : increased energy of k th waveform.

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Transmit Beamspace Design

The transmit weight matrix C is designed based on maximiz-ing the ratio

"k !=

T="

66cHk a(#)!k (t)

662d#dt=

T= %#% |c

Hk a(#)!k (t)|2d#dt

=cH

k>=

" a(#)aH(#)d#?ck= %

#% |cHk a(#)|2d#

=cH

k Ack

2(cHk ck

(7)

A !=" a(#)aH(#)d# is a positive definite matrix.

The maximization of the expression (7) is equivalent to max-imizing its numerator while fixing its denominator.

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Transmit Beamspace Design (Cont’d)

One way to insure that c1 *= c2 . . . *= cK is to impose theconstraint CHC = I.The maximization of {"k}K

k=1 subject to the constraint CHC =I yields

C = [u1,u2, . . . ,uK ]

{ui}Ki=1 are K principal eigenvectors of A.

It is worth noting that the use of principle eigenvectors fortransmit beamforming has been used by [Frazer, et. al.,2007] to minimize the reactive power at the transmitter.

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Transmit Beamspace Based MUSIC

The virtual data model (6) can be rewritten as

y($) =

%EK

V#($) + zK ($)

#($)! ['1($), . . . ,'L($)]T

V! [v(#1), . . . ,v(#L)]v(#) =

>CHa(#)

?# b(#).

The KN " KN transmit energy focusing based covariancematrix is then given by

R ! E9

y($)yH($):=

EK

VSVH + %2z IKN

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Transmit Beamspace Based MUSIC (Cont’d)

The eigendecomposition of R can be written asR = Es#sEH

s + En#nEHn .

The transmit energy focusing based spectral-MUSIC estima-tor can be expressed as

f (#) =vH(#)v(#)

vH(#)Qv(#)Q = EnEH

n = I % EsEHs is the projection matrix onto the noise

subspace.Substituting v(#) into (31), we obtain

f (#) =NaH(#)CCHa(#)

@>CHa(#)

?# b(#)

AHQ@>

CHa(#)?# b(#)

A .

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Cramer-Rao BoundThe CRB for estimating the DOAs can be expressed as

CRB(#) =%2

z K2QE

9Re

-DHP%

V D + GT.:#1

P%V ! V(VHV)#1VH , G ! (SVHR#1VS), D ! [d(#1), . . . ,d(#L)]

d(#) =d@(CHa(#))# b(#)

A

d#= (CHa&(#))# b(#) + (CHa(#))# b&(#).

It can also be shown that if CK+1 = [CK , cK+1] (cK+1 is anon-principal eigenvector) is used, then we have

CRBK+1 > CRBK .

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Simulation Results

ULA, # = )/2, M = 10, and " = [%10$ 10$].Two types of receivers (i) a single receive antenna; (ii) anarbitrary linear receive array of N = 10 antennas; randomlocations.Two targets are located at directions 2$ and 4$.Fixed total transmit energy E = M.Targets are assumed to be resolved if

|#i % #i | !|#2 % #1|

2, i = 1, 2.

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Simulation Results (Cont’d)

RMSE versus SNR

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Probability of source resolution

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CRB versus number of transmit beams K

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Transmit Beamspace Signal Model

Assume L targets are located within spatial sector ".Each transmit subarray is taken as the whole transmit array,i.e., all K subarrays are identical.Let W(#) = [w1, . . . ,wK ] be the M " K transmit beamspacematrix.The M " 1 weight vector wk can be designed to focus theenergy of the k th waveform within $.The N " 1 received vector

xbeam(t , q)=%

EK

L,

l=1

"l(q)b(#l)-

WHa(#l).T

!(t)+z(t , q).

!(t) ! [!1(t), . . . ,!K (t)]T : K " 1 waveform vector.

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Transmit Beamspace Signal Model (Cont’d)

Match-filtering to the K transmitted waveforms

yk (q) =

/

Txbeam(t , q)!"

k (t)dt , k = 1, . . . ,K

=

%EK

L,

l=1

"l(q)-

wHk a(#l)

.b(#l) + zk (q). (8)

Rewrite (8) as

yk (q) = Bk$(q) + zk (q). (9)

Bk ! [b(#1), . . . ,b(#L)]$k$k ! diag

BwH

k a(#1), . . . ,wHk a(#L)

C

$(q) ! ["1(q), . . . ,"L(q)]T .The form (9) enjoys rotational invariance properties

Bk = Bj$#1j $k , k , j = 1, . . . ,K .

Therefore, PARAFAC/ESPRIT can be applied to (8).38 / 108


Transmit Beamspace Based ESPRIT

When K = 2, i.e., two transmit beams

B2 = B1$

$ ! diag

7wH

2 a(#1)

wH1 a(#1)

, . . . ,wH

2 a(#L)

wH1 a(#L)

8. (10)

Rewrite (10) as

$ = diag9

A(#1)e!!(#1), . . . ,A(#L)e!!(#L):

A(#) = |wH2 a(#)/wH

1 a(#)|, !(#) = angleB

wH2 a(#)/wH

1 a(#)C

.ESPRIT-based techniques can estimate $.

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Special Case: ULA Transmit Array

Assuming w1 = [wT 0]T and w2 = [0 wT ]T , then

wHa2(#) = wHa1(#)e#! 2!d" sin #

a1(#): contains first M % 1 of a(#),a2(#): contains last M % 1 of a(#).In this case,

$ = diag9

e#! 2!d" sin #1 , . . . ,e#! 2!d

" sin #L:.

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Simulation Results

M = 10 transmit antennas spaced half a wavelength apart.N = 10 receive antenna (arbitrary array)Two target located at #1 = 1$ and #2 = 3$

The sector of interest is [%5$ 5$].Fixed total transmit energy E = M.Two orthogonal transmit waveforms are used.w = [%0.5623 % 0.5076 % 0.4358 % 0.3501 % 0.2542%0.1524 % 0.0490 0.0512 0.1441]T .Targets are assumed to be resolved if

|#i % #i | !|#2 % #1|

2, i = 1, 2.

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2D Arrays

Sparse 2D transmit arrays are commonly used in practice toreduce the cost.Sparse 2D arrays are also used to realize dual-band radarsystems.Sparse arrays do not straightforwardly enable the use ofcomputationally efficient direction finding algorithms.

A method for mapping an arbitrary transmit array into anarray with virtually uniform structure enables

SNR improvement due to 2D transmit beamforming gain.Root-MUSIC for computationally efficient target localization.

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Signal Model and Problem Formulation

Consider a MIMO radar with planar transmit and receivearrays of M and N antennas, respectively.The transmit array can be a non-uniform array or a sparsearray.The M " 1 transmit array steering vector can be representedas

a(#,!) =De#j2%µT (#,&)p1 , . . . ,e#j2%µT (#,&)pM

ET

# and !: Elevation and Azimuth angles, respectivelypm ! [xm ym]T , m = 1, . . . ,M: Locations of the transmitantennasµ(#,!) = [sin # cos! sin # sin!]T : The propagation vector.

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Signal Model and Problem Formulation (Cont’d)Let C = [c1, . . . ,cQ] be the M " Q interpolation matrix(Q ! M).Let d(#,!) be the Q " 1 interpolated array steering vector

CHa(#,!) ( d(#,!) # , ", ! , !

The signal radiated towards a hypothetical spatial location(#,!) is

*(t , #,!)=dT (#,!)s(t) =Q,

i=1

-cH

i a(#,!).

si(t)

s(t) = [s1(t), . . . , sQ(t)]: Q " 1 vector of orthogonal wave-forms.

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Signal Model and Problem Formulation (Cont’d)Assuming L targets, the receive observation vector is

x(t , $) =L,

l=1

"l($)-

dT (#l ,!l)s(t).

b(#l ,!l) + z(t , $)

$ : Radar pulse number"l($): Reflection coefficient associated with l-th targetb(#,!): N " 1 steering vector of the receive arrayz(t , $): N " 1 additive noise vector.Matched-filtering x(t , $) to si(t) yields

yi($) =L,

l=1

"l($)

;cH

i a(#l ,!l)

<

! "# $Transmit Processing Gain

b(#l ,!l) + zi($), i = 1, . . . ,Q

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LS-based Transmit Array InterpolationThe interpolation matrix C can be computed as the LS solu-tion to

CHA = A (11)

A = [a(#1,!1), . . . ,a(#K",!1), . . . ,a(#K"

,!K#)] (size : M"K#K&)

A = [a(#1,!1), . . . , a(#K",!1), . . . , a(#K"

,!K#)] (size : M"K#K&)

#k , ", k = 1, . . . ,K#: Angular grid that approximates thesector "!k , !, k = 1, . . . ,K&: Angular grid that approximates thesector !.Given that K#K& ) M, the LS solution to (11) can be givenas

C =-

AAH.#1

AAH

This solution exhibits very high sidelobe levels resulting inwasting the transmit power within the out-of-sector regions!

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Transmit Interpolation Matrix Design

Minimize the maximum difference between the mapped arraysteering vector and the desired one within a certain sector(",!) while keeping the sidelobe level bounded by someconstant.The interpolation matrix design (convex) problem:

minC

max#k ,&k!

FFFCHa(#k ,!k !)% d(#k ,!k !)FFF

1(12)

#k , ", k = 1, . . . ,K#, !k ! , !, k & = 1, . . . ,K&

subject toFFFCHa(#n,!n!)

FFF1! +,

#n , ", n = 1, . . . ,N#, !n! , !, n& = 1, . . . ,N&

+ is a positive number of user choice used to upper-boundthe sidelobe level.

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Transmit Interpolation Matrix Design (Cont’d)

Alternative interpolation matrix design (convex) problem:

minC

max#n,&n!

FFFCHa(#n,!n!)FFF

1(13)

#n , ", n = 1, . . . ,N#, !n! , !, n& = 1, . . . ,N&

subject toFFFCHa(#k ,!k !)% d(#k ,!k !)

FFF1! #

#k , ", k = 1, . . . ,K#, !k ! , !, k & = 1, . . . ,K&

# is a positive number of user choice used to control thedeviation of the interpolated array from the desired one.Note: Other requirements can also be enforced by imposingadditional constraints in (12) and (13). Spectral Constraints[Rowe, Stoica, Li’14]

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Root-MUSIC Based Localization

Assume L-shaped desired array with half wavelength inter-element spacing d(#,!) = [uT (#,!) vT (#,!)]T

u(#,!)=D1, e#j% sin # cos&, . . . ,e#j%(Md#1) sin # cos&

ET

v(#,!)=D1, e#j% sin # sin&, . . . ,e#j%(Nd#1) sin # sin&

ET

Md : Number of desired elements on x-axisNd : Number of desired elements on y-axisSolving either (1) or (2) yields

CHa(#,!) - [uT (#,!) vT (#,!)]T , # , ", ! , !

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Root-MUSIC Based Localization (Cont’d)

Construct the Md " N data matrix

Yu($) = [y1($), . . . ,yMd ($)]T

-L,

l=1

"l($)u(#,!)bT (#l ,!l) + Zu($)

Build the Md " Md covariance matrix

Ru =Ts,

"=1

Yu($)YHu ($)

Apply root-MUSIC to obtain the estimates

,l = sin #l cos!l , l = 1, . . . ,L

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Similarly, construct the Nd " N data matrix

Yv ($) = [yMd+1($), . . . ,yMd+Nd ($)]T

-L,

l=1

"l($)v(#,!)bT (#l ,!l) + Zv ($)

Build the Nd " Nd covariance matrix

Rv =Ts,

"=1

Yv ($)YHv ($)

Apply root-MUSIC to obtain the estimates

-l = sin #l sin!l , l = 1, . . . ,L

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The ,l and -l estimates can be further arranged in the form

.l = ,l + j-l , l = 1, . . . ,L

The estimates of the elevation angles are obtained as

#l = sin#1(|.l |), l = 1, . . . ,L

The estimates of the azimuth angles are obtained as

!l = "(.l), l = 1, . . . ,L

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Simulation Results

M = 64 transmit antennas; x- and y-components of theirpositions be drawn randomly from the interval [0, 8)]Transmit sector: " = [30$, 40$] and ! = [100$, 110$]

L-shaped array: Md = 5 equally spaced along the x-axis;Nd = 5 equally spaced along the y-axisUnknown target direction # = 35$ and ! = 105$

C is designed using (13) for # = 0.1, 0.02, and 0.001N = 16 elements receive array (locations drawn randomlyfrom [0, 10)])

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Transmit beampattern for # = 0.1

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Probability of in-sector detection versus SNR.

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RMSE versus SNR.

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ESPRIT-based 2D DOA Estimation

Without loss of generality, we choose the desired array to betwo perpendicular linear subarrays of two elements each.The virtual locations of the elements of the two subarraysare [x1, 0]T , [x2, 0]T , [0, y1]

T , and [0, y2]T (x1, x2, y1, and

y2 measured in wavelength).Also choose the desired a(#,!) to take the following format

a(#,!) =

0

112

e#j2%x1 sin #

e#j2%x2 sin #

e#j2%y1 sin&

e#j2%y2 sin&

3

445, # , ", ! , !.

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ESPRIT-based 2D DOA Estimation (Cont’d)

The matched filtered virtual data at the receiver simplifies to

y1($) -L,

l=1

"l($)e#j2%x1 sin #l b(#l ,!l) + z1($)

y2($) -L,

l=1

"l($)e#j2%x2 sin #l b(#l ,!l) + z2($)

y3($) -L,

l=1

"l($)e#j2%y1 sin&l b(#l ,!l) + z3($)

y4($) -L,

l=1

"l($)e#j2%y2 sin&l b(#l ,!l) + z4($)

y1 and y2 enjoy rotational invariance that enables estimating#l , l = 1, . . . ,L.y3 and y4 enjoy rotational invariance that enables estimating!l , l = 1, . . . ,L.

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Simulation Results

8 " 8 non-uniform rectangular transmit array" = [30$, 40$] and ! = [95$, 105$]

x1 = y1 = )/2 and x2 = y2 = ) () is the wavelength)Two targets located !1 = 98$, !2 = 101$, #1 = 33$ and#2 = 37$

C is designed using (13)–(14) using # = 0.1N = 16 elements receive array (locations drawn randomlyfrom [0, 2)])

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Normalized transmit beampattern

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Left: Phase rotation between first and second elements of theinterpolated arrays. Right: Phase rotation between third and

forth elements of the interpolated arrays66 / 108



RMSE versus SNR

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Capability of Jammers Suppression

Jammers suppression is one of the most important issuesin radar signal processing.MIMO radar with colocated antennas shows advantagesover phased-array (PA) radar, and provides designers withmore opportunities to achieve multiple goals.Capability of efficient suppression on powerful jammers forMIMO radar has not been studied in previous works.Spatial processing techniques (with reduced dimension(RD)) such as beamspace processing or beamforming tech-niques significantly reduce the computational complexity.

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Contents

Three RD beamspace designs with robustness or adap-tiveness are proposed.Reasonable tradeoffs among desired in-sector source pre-servation, powerful in-sector jammer suppression, and out-of-sector interference attenuation are made when designingbeamspace matrices.Two robust beamformers are designed for known/unknownin-sector jammers suppression.Efficient source power estimates in the context of powerfuljammers and non-ideal factors are derived.Capability of efficient in-sector jammers suppression usingthese designs is shown to be unique in MIMO radar.

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Background

Jamming signals take the form of high-power transmissionthat aim at impairing the receive system.Terrain-scattered jamming occurs when the high-power jam-mer transmits its energy to ground, and it reflects the energyin a dispersive manner.Jamming appears at the receive array as distributed source.Jammers suppression becomes more challenging when jam-ming impinges on the receive array from the same directionas the desired target.Situation worsens even more since the powerful jammersare generally unknown.

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Signal Model

L targets including the desired and interfering sources, Jpresumed powerful jammersThe received signal can be modeled as

y($) =L,

l=1

'l($)v(#l) +J,

j=1

"j($)v>#j?+ ~z($)

where!"

#$v(#l) ! a(#l)# b(#l) and

!"

#$v(#j) ! 1M # b(#j) .

Jammer does not originate from the transmit array, therefore,does not depend on the transmit array steering vector.

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RD Beamspace Processing

RD beamspace matrix B of size MN " D (D ' MN) withtransformation

y($) = BHy($).

Covariance matrix:

Eigendecomposition:

Ry ! EB

y($)yH($)C= BHRyB

Ry = Es#sEHs + En#nEH

n

Beamspace spectral-MUSIC DOA estimator

f (#) =vH(#)BBHv(#)

vH(#)BQBHv(#)

Q ! EnEHn = ID % EsEs is the projection matrix.

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Assumptions

Desired targets are located within a known (or pre-estimated)angular sector ".Powerful jamming sources are also present within the sector-of-interest ", and can even have the same spatial directionsas targets.Interfering sources are present in the out-of-sector spatialarea ".Generally, all the possible in-sector jamming and out-of-sector interfering sources are unknown.

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Beamspace Design I

Build a quiescent response beamspace matrix Bq usingspheroidal sequences based methods. " using spheroidalsequences based methods.Upper-bound the acceptable difference between the desiredbeamspace matrix B and the quiescent one Bq while maxi-mizing the worst-case in-sector jammers suppression.Maintain out-of-sector sidelobes below a certain level.

minB

maxi

FFFBH v(#i)FFF, #i , ", i = 1, . . . ,Q

s.t.FFB % Bq

FFF ! /

FFFBHv>#k?FFF ! +, #k , ", k = 1, . . . ,K .

(14)

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Beamspace Design II

Build a quiescent beamspace matrix Bq.Minimize the difference between the desired and quiescentresponse beamspace matrices while keeping the in-sectorjammers suppression higher than a certain desired level.If needed, keep the out-of-sector interference attenuation atan acceptable level.

minB

FFB % BqFF

F

s.t.FFFBH v(#i)

FFF ! 0, #i , ", i = 1, . . . ,QFFFBHv

>#k?FFF ! +, #k , ", k = 1, . . . ,K .

(15)

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Beamspace Design III

Data-adaptive approach is meaningful especially when jam-mers and/or interfering sources are varying.Build a quiescent beamspace matrix Bq.Minimize output power of y($) while maintaining beamspacematrix difference, in-sector jammers suppression, and out-of-sector interference attenuation below desired levels.

minB

tr9

BHRyB:

s.t.FFB % Bq

FFF ! /

FFFBH v(#i)FFF ! 0, #i , ", i = 1, . . . ,Q

FFFBHv>#k?FFF ! +, #k , ", k = 1, . . . ,K .

(16)

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Technical Details

Feasibility

(14) and (15) are always feasible for reasonable 0 and +.

Feasibility of (16) is guaranteed if / ) /min (minimum value of&B % Bq&F by solving (15)) is used for fixed 0 and +.

Beamspace dimension

Depends on the width of ", no smaller than the number ofprincipal eigenvalues of A !

=" vH(#)v(#)d#.

No smaller than the number of desired targets for MUSIC-based DOA estimation.

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Robust Beamforming I

Case I: Known Jammer DirectionsMaintain a distortionless response towards the desired targetdirection #t .Enforce deep nulls towards the spatial directions of the jam-mers.

minw

wHRw

s.t. wHv(#t) = 1

wH v>#j?= 0, j = 1, . . . , J

R: MN " MN covariance matrix of interference plus jammerand noise.

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Robust Beamforming II

Case II: Unknown Jammer DirectionsEnforce deep nulls towards all the possible directions of thejammers and, if needed, the out-of-sector interfering sourceattenuation should be kept to an acceptable level.

minw

wHRw

s.t. wHv(#t) = 1666wH v(#i)

666 ! 0, #i , ", i = 1, . . . ,Q666wHv

>#k?666 ! +, #k , ", k = 1, . . . ,K .

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Source Power Estimation

Performance degrades when array calibration errors and mis-matches between the presumed and actual target steeringvectors are present.Robust beamforming design II can be approximated by thefollowing strengthened optimization problem

minw

wHRw

s.t. wH v(#t) = 1

&wH V&2 ! 0

where V ! [v(#1), . . . , v(#Q)], 0 ! Q02, and &v(#)%v(#)&2 !1.

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Source Power Estimation (Cont’d)

Closed-form solutions

I. If vH (#t )R!1RVR!1v(#t )

[vH (#t )R!1v(#t )]2 ! 0, then w = R!1v(#t )

vH (#t )R!1vH (#t ).

II. If vH (#t )R!1RVR!1v(#t )

[vH (#t )R!1v(#t )]2 > 0, then w =

(R+&RV)!1v(#t )

v(#t )H(R+&RV)

!1v(#t ).

Source power estimate can be derived as

%20 =

v(#t)H>R + )RV

?#1R>R + )RV

?#1v(#t)Dv(#t)

H>R + )RV?#1v(#t)

E2 (17)

which is valid for both conditions () = 0 for the first one),and RV ! VVH .

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Simulation Results of Beamspace Designs

ULA, dT = dR = )/2, M = 16, N = 8, and " = [10$, 25$].Target DOAs #t = 16.5$ and 18.5$.Interfering source DOAs # = %35$, %20$,%5$, and 50$,respectively.SNR = 0 dB, INR = 40 dB, and JNR = 50 dB.D = 7, + = 0.2, 0 = 0.1, and / = 1.467.Targets are assumed to be resolved if

|#i % #i | !|#2 % #1|

2, i = 1, 2.

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Simulation Results of Beamspace Designs (Cont’d)

!80 !60 !40 !20 0 20 40 60 80!100

!90

!80

!70

!60

!50

!40

!30

!20

!10

0

Angle(Degrees)

Beam

space

attenuatio

n(d

B)

Spheroidal sequences based design

Adaptive beamspace design !!!"Jammers

Beamspace attenuation versus angle. Uniformly distributedjammers spaced 1$ apart from each other are present.

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Simulation Results of Beamspace Designs (Cont’d)

0 10 20 30 40 50 6010

!4

10!3

10!2

10!1

100

101

102

SNR(dB)

RM

SE

(De

gre

es)

Elementspace MUSIC

Spheroidal sequences based design

Beamspace design #Beamspace design !!Beamspace design !!!

(a) RMSEs of DOA estimation versusSNR.

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR(dB)

Prob

abilit

ies

of s

ourc

es re

solu

tion

Elementspace MUSICSpheroidal sequences based designBeamspace design !Beamspace design !!Beamspace design !!!

(b) Probabilities of source resolutionversus SNR.

DOA estimation performance versus SNR. 5 jammers locatedbetween 15.5$ and 19.5$ are present.

84 / 108


Simulation Results of Robust Beamforming

ULA, dT = dR = )/2, M = 10, N = 10, and " = [10$, 25$].Target DOA #t = 18$.Interfering sources DOAs # = %35$, %20$,0$, and 50$,respectively.SNR = 5 dB, INR = 40 dB, and JNR = 50 dB.0 = 0.0001.

85 / 108


Simulation Results of Robust Beamforming (Cont’d)

!80 !60 !40 !20 0 20 40 60 80!120

!100

!80

!60

!40

!20

0

20

40

Angle (Degrees)

Beam

pattern

(dB

)

Robust Design ! (Target)"Conventional Beamform$% (Target)

Robust Design ! (Jammer)"Conventional Beamform$% (Jammer)

(c) Beampatterns versus angles.

!80 !60 !40 !20 0 20 40 60 80

!20

!10

0

10

20

30

40

50

Angle (Degrees)

Pow

er

est

imate

(dB

)

Robust&'()*+($ConventionalBeamform$%MVDRBeamform$%

(d) Power estimates versus angles.The first example. Uniformly distributed jammers spaced 1$

apart from each other are present.

86 / 108


Simulation Results of Robust Beamforming (Cont’d)

!80 !60 !40 !20 0 20 40 60 80!120

!100

!80

!60

!40

!20

0

20

40

Angle (Degrees)

Beam

pattern

(dB

)

Robust Design !! (Target)"Conventional Beamform$% (Target)

Robust Design !! (Jammer)"Conventional Beamform$% (Jammer)

(e) Beampatterns versus angles.

!80 !60 !40 !20 0 20 40 60 80

!20

!10

0

10

20

30

40

50

Angle (Degrees)

Pow

er

est

imate

(dB

)

Robust&'()*+($ConventionalBeamform$%MVDRBeamform$%

(f) Power estimates versus angles.The second example. Three jamming signals with the DOAs

#j = 15$, 18$, and 21$, respectively, are present.

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Ambiguity Function of the TB-Based MIMO Radar

Ambiguity function (AF) serves as a fundamental tool toevaluate radar waveform or resolution performance.Transmit beamspace (TB)-based MIMO radar focuses en-ergy of multiple transmitted waveforms within a certain spa-tial sector where a target is likely present via TB designtechniques.Essence of the TB-based MIMO radar is to achieve im-proved SNR gain and increased aperture simultaneously.Superior DOA estimation performance in a wide range ofSNRs can be achieved in the TB-based MIMO radar.

88 / 108


Contents

AF for the TB-based MIMO radar for the case of far-fieldtargets and narrow-band waveforms is derived.Equivalent transmit phase centers are introduced in the AFdefinition.It serves as a generalized AF form for the existing radar con-figurations including phased-array (PA) radar and traditionalMIMO radar (with subarrays).Relationships between the defined TB-based MIMO radarAF and the previous AF works are established.

89 / 108


Background

Woodward’s AF serves as the groundbreaking work on AF.Traditional MIMO radar AF becomes invalid for TB-basedMIMO setup since TB processing allows non-orthogonal orcorrelated waveforms to be emitted at each antenna.Waveform correlation matrix design for desired transmitbeampattern can be simplified to TB matrix design in tradi-tional MIMO radar.In-depth study on TB-based MIMO radar AF may also pro-vide insight into facilitating the clutter/interference mitigationin airborne MIMO radar system using TB design.

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Signal Model

K (in general,K ! M) initially orthogonal waveforms areemitted.K transmit beams that fully cover the pre-estimated spatialsector % are formed using TB matrix C ! [c1, . . . ,cK ].Signal modelSignal radiated via the k th beam:

sk (t) =%

EK

cTk a(#)!k (t), k = 1, . . . ,K .

Signal radiated from the mth antenna:

sm(t) =%

EK

K,

k=1

cmk!k (t), m = 1, . . . ,M.

91 / 108


Signal Model (Cont’d)

Let the received signal at the j th antenna be rj(t ,").Matched filtered noise-free signal component w.r.t. the i thwaveform is

r &ji>","&? =

/rj(t ,")!"

i>t ,"&?dt

=

%EK

M,

m=1

K,

k=1

'mj

/cmk!k

>t % $mj(p)

?!"

i>t % $q(i)j

>p&??

" expB%j2($mj(p)

>fc + fmj(")

?C

" expB

j2($q(i)j>p&?>fc + fq(i)j

>"&??C

" expB

j2(>fmj(")% fq(i)j

>"&??t

Cdt .

92 / 108


AF Definition

Define the AF as the square of coherent summation of allnoise-free matched filter output pairs, i.e.,

.(","!) !

666666

N,

j=1

K,

i=1

r &ji>","!?

666666

2

=

666666

N,

j=1

K,

i=1

M,

m=1

'mj [R]mi(","!,C, j)exp{%j2($mj(p)

"(fc + fmj("))}exp{j2($q(i)j(p&)(fc + fq(i)j("!))}

666666

2

93 / 108


AF Definition (Cont’d)

[R]mi(","!,C, j) !%

EK

K,

k=1

cmk

/!k (t % $mj(p))

" !"i (t % $q(i)j(p&))exp{j2((fmj(")% fq(i)j("!))t}dt

!k : the k th orthogonal waveform.fc : radar operating frequency.fmj : Doppler shift due to the (m, j)th transmit-receive

channel.$mj : two-way time delay due to the (m, j)th channel.q(i): the i th equivalent transmit phase center.

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AF Illustration

Defined AF is composed of summation terms.

Each term contains three components:Complex reflection coefficientMatch-filtered component standing for waveform correlationExponential terms standing for phase shift information due torelative position and target motion w.r.t. array geometry.

Defined AF means that the mth antenna emits a signalconsisting of all K initially orthogonal waveforms windowedby elements of the mth row in C.

95 / 108


AF Simplification

Neglect complex coefficients (with constant contributions).Simplify AF definition as

.>","!? =

666aHR (")aR

>"&?

6662666aH

T (")R(#$,#fd ,C)aTE>"!?

6662

aT, aR, aTE are the transmit, the receive, and the equivalenttransmit array steering vectors, respectively.

R derives from R with #$ and #fd being used.

@RA

mi(#$,#fd ,C) =

%EK

K,

k=1

cmk

/!k (t)!"

i (t %#$)exp{j2(#fd t}dt

96 / 108


Relationships With Other AFs

Express the simplified AF using Woodward’s AF:

.>","!? = E

K

666aHR (")aR

>"&?

6662666aH

T (")C%(#$,#fd)aTE>"!?

6662.

Traditional MIMO Radar: K = M, C = IM , aTE = aT(uniform subarrays: C is block diagonal, choose subarraycenters for aTE)

.MIMO>","!? = E

M

666aHR (")aR

>"&?

6662666aH

T (")%(#$,#fd)aT>"!?

6662.

PA Radar: K = 1, C = wM (beamforming weights), aTE = 1

.PA>","!? = E

666aHR (")aR

>"&?

6662666aH

T (")wM.(#$,#fd)6662.

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Weak and Strong “Clear Region” Bounds

Worst-Case Bound

C I(A) ! 4VKN2K

|aHR (")aR("!)|2

VK % 42 (18)

Best-Case Bound

CII(A) ! 4VKN2

|aHR (")aR("!)|2

VK % 42 (19)

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Simulation Results

ULA, dT = dR = )/2, M = 8, N = 8, and E = M.Array locates on x-axis.Polyphase-coded waveforms with code length being 256.T = 10 ms, BT = 128, and fs = 2B.Single pulse.Two targets locate on y -axis.

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Difference between the defined TB-based MIMO radar AF andthe squared-summation-form traditional MIMO radar AF, 8

polyphase-coded waveforms are employed.

100 / 108



TB-based MIMO radar AF, 4 polyphase-coded waveforms areemployed.

101 / 108


References

A. Hassanien and S. A. Vorobyov, “Phased-MIMO radar:A tradeoff between phased-array and MIMO radars, IEEETrans. Signal Process., vol. 58, no. 6, pp. 3137–3151,June 2010.A. Hassanien and S. A. Vorobyov, “Transmit energy focusingfor DOA estimation in MIMO radar with colocated antennas,”IEEE Trans. Signal Process., vol. 59, no. 6, pp. 2669-2682,June 2011.A. Hassanien, S. A. Vorobyov, and A. B. Gershman, “Mov-ing target parameters estimation in non-coherent MIMOradar systems,” IEEE Trans. Signal Process., vol. 60, no. 5,pp. 2354-2361, May 2012.

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References (Cont’d)

A. Khabbazibasmenj, A. Hassanien, S. A. Vorobyov, andM. W. Morency, “Efficient transmit beamspace design forsearch-free based DOA estimation in MIMO radar,” IEEETrans. Signal Process., vol. 62, no. 6, pp. 1490-1500, Mar. 2014.A. Hassanien and S. A. Vorobyov, “Direction finding for MIMOradar with colocated antennas using transmit beamspacepreprocessing,” in Proc. IEEE CAMSAP, Dutch Antilles,Dec. 2009, pp. 181–184.A. Hassanien and S. A. Vorobyov, “Transmit/receive beam-forming for MIMO radar with colocated antennas," in Proc.34th IEEE ICASSP, Taiwan, Apr. 2009, pp. 2089-2092.

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A. Hassanien and S. A. Vorobyov, “Why the phased-MIMOradar outperforms the phased-array and MIMO radars,” inProc. 18th EUSIPCO’10, Denmark, Aug. 2010, pp. 1234-1238.A. Hassanien, S. A. Vorobyov, A. B. Gershman, and M. Rub-samen, “Estimating the parameters of a moving target inMIMO radar with widely separated antennas,” in Proc. IEEESAM Worshop, Israel, Oct. 2010, pp. 57-60.A. Hassanien and S. A. Vorobyov, “Subspace-based direc-tion finding using transmit energy focussing in MIMO radarwith colocated antennas,” in Proc. IEEE ICASSP, CzechRepublic, May 2011, pp. 2788-2791.

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A. Khabbazibasmenj, S. A. Vorobyov, A. Hassanien, andM. W. Morency, “Transmit beamspace design for directionfinding in colocated MIMO radar with arbitrary receive arrayand even number of waveforms,” in Proc. Asilomar, USA,Nov. 2012, pp. 1307-1311.A. Hassanien, M. W. Morency, A. Khabbazibasmenj,S. A. Vorobyov, J.-Y. Park, and S.-J. Kim, “Two-dimensionaltransmit beamforming for MIMO radar with sparse symmetricarrays,” in Proc. IEEE Radar Conf., Canada, Apr. 2013.A. Hassanien, S. A. Vorobyov, and J.-Y. Park, “Joint trans-mit array interpolation and transmit beamforming for sourcelocalization in MIMO radar with arbitrary arrays,” in Proc.ICASSP, Canada, May 2013, pp. 4139-4143.

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A. Hassanien, S. A. Vorobyov, Y.-S. Yoon, and J.-Y. Park,“Root-MUSIC-based source localization using transmit arrayinterpolation in MIMO radar with arbitrary planar array,” inProc. CAMSAP, Saint Martin, Dec. 2013, pp. 396-399.Y. Li, S. A. Vorobyov, and A. Hassanien, “MIMO radar ca-pability on powerful jammers suppression,” in Proc. IEEEICASSP, Italy, May 2014, pp. 5297–5301.Y. Li, S. A. Vorobyov, and A. Hassanien, “Robust beamform-ing for jammers suppression in MIMO radar,” in Proc. IEEERadar Conf., USA, May 2014.Y. Li, S. A. Vorobyov, and V. Koivunen, “Generalized ambigu-ity function for the MIMO radar with correlated waveforms,”in Proc. IEEE ICASSP, Italy, May 2014, pp. 5322–5326.

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Contributors

1Sergiy A. Vorobyov Aalto University2Aboulnasr Hassanien University of Alberta3Yongzhe Li Aalto University4Arash Khabbazilasmenj University of Western Ontario5Matthew W. Morency University of Alberta6Visa Koivunen Aalto University7 Samsung-Thales Co., Ltd.

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Trade-Offs in MIMO Radar with widely-spaced antennas

Marco Lops

Outline • MIMO Radar with widely spaced antennas:

Angular diversity and its consequences; • Signal models, diversity, Space-Time Coding; • Detection through MIMOs: cost functions, figures

of merit and tradeoffs; • Robust waveform design; • Localization through MIMO radars: should we

really maximize angle diversity? • Conclusions

MIMO Radar • Decentralized systems: multiple receivers and a

fusion center collecting either raw data or a quantized version thereof;

• Multi-site Radars: systems joining in the network, with no real transmitter cooperation;

• Communication-theory-related MIMO: Multiple transmitters, cooperating to form the radiated waveforms and operating under power constraints, along with multiple receivers, cooperating in order to make final decisions.

Aspect angles

Transmit/Receive Diversity

Receive diversity: Transmit diversity:

Interpretation: Receive diversity A target whose size in the receive array alignment direction is V’ becomes an aperture antenna whose beam has width λ/V’. The scattering towards two elements spaced d’ apart is independent if they are not seen as belonging to the same cell. Such a cell at distance R’ has size R’ λ/V’.

Remark: The interpretation of transmit diversity is the same.

Consequence The same target may not offer the same amount of angle diversity over time. Indeed, the number of independent paths turns out to depend on the target orientation (i.e., on its size along the transmit array direction) and on its range. Additionally, the same target may offer: Full transmit and/or receive diversity; Full transmit diversity and partial receive diversity (or vice-versa); Partial (or no) diversity (transmit and/or receive). The issue of designing waveforms which are robust with respect to the amount of real diversity is central in MIMO radar. For a physical (electromagetic) study of the relationship between target scattering and available d.o.f. see, e.g., M. D. Migliore, “Some physical limitations in the performance of statistical multiple-input multiple-output RADARs”, IET Microw. Antennas and Propagations, vol. 2, N. 8

Angular diversity: models

• Full transmit diversity: α1,T, α2,T, ... αM,T are statistically independent;

• Receive diversity: α(1)T,1, α(1)

T,2, ... α(1)T,L, are

statistically independent, and so are α(i)T,1 , ... α(i)

T,L for i=2,...,M

• Full diversity: both transmit and receive diversity; • Notice that the amount of diversity varies with V,

V’, the target distance, and the array elements spacing.

Full diversity

The matrix α=(α1,α2) has rank 2 almost surely.

Signal models

About the delays

• Resolvable paths: the delays are all resolvable, which means that the available transmit bandwidth is large enough as to allow resolvability for most of the distances.

• Un-resolvable paths: the radar is “narrow-band”, whereby the target appears in one and the same range cell.

• The most interesting situation to study the trade-offs is the latter, as will be shown later.

Un-resolvable paths: conditions

Resolvable paths

In this case the following “orthogonality” is advocated

This requires either extremely large bandwidths, or extremely wide spacing or radars operating on disjoint spectra. It is reminiscent more of frequency diversity than of space diversity.

Choosing the waveforms: Space-Time coding

A viable way to design the waveforms is to set

Example#1: Pulsed STC radar

Example 2: Sophisticated STC Radar

The STC Matrix

The key parameter is the STC matrix

It may be worth noticing that the simplest case is to make c real.

Key parameters • Transmitted energy, encapsulated in trace(CCH): it

represents an overall constraint that applies to the transmitted waveforms and allows performing comparisons and eliciting trade-offs;

• Rank(C), i.e. the rank of the STC matrix, encapsulating the transmit policy: for both resolvable - where it equals the number of active antennas- and un-resolvable paths it determines how many degrees of freedom we choose to exploit a priori.

Degrees of freedom

The matrix C defines the degrees of freedom, which are obviously

The reason is that, for fat matrix, the code-words are no longer linearly independent. As a consequence, for un-resolvable paths the signal space is obviously of reduced dimension. From now on we consider the far more challenging (as far as trade-offs are concerned) situation of un-resolvable paths, since most of the conclusions, at least for target detection, carry over to the case of resolvable paths.

Target detection: signal model

The signal received at receive antenna i, ri ∈ C N , reads:

Where αi is the scattering towards antenna i (and thus has dimension M) while wi represents the impinging disturbance, which is assumed Gaussian with covariance matrix

Memo: symbol meaning

Generalized Likelihood Ratio Test (GLRT, scattering as nuisance)

If C is full rank, the number of degrees of freedom is Lmin(N,M) since:

Detector structure (un-resolvable paths)

Interesting Special cases

White noise, N=M and orthogonal coding:

GLRT: (Incoherent integrator on ML paths)

Rank-one coding (e.g., C=11T, corresponds to an incoherent power multiplexer)

GLRT:

No transmit diversity, only receive diversity, since whatever is transmitted collapses onto the unique direction of the signal space.

(Incoherent integrator on L coherent integrators)

Remark: resolvable paths For resolvable paths, rank-deficient coding would correspond to activating just a subset of the transmit antennas. Once an antenna is activated, it necessarily introduces L diversity paths if the “orthogonality hypothesis under any delay” applies: otherwise stated, if all of the antennas are activated we generate LM diversity paths, but by taking a much larger bandwidth. The optimum detector is obviously an incoherent energy integrator. Thus, un-resolvable paths with full-rank coding achieve the same detection performance as resolvable paths with M active antennas. Question: can a MIMO radar with widely spaced antennas be considered as a mechanism achieving spatial diversity? Or – rather – it enjoys a form a frequency diversity, possibly combined with angle diversity if the transmitters operate on non-disjoint bandwidths, arising from the wide-band transmission format? This is debatable.

Effect of diversity on detection performance

Assume rank-δ coding under un-resolvable paths (the case of resolvable paths with δ transmitters is similar). The detection perfomance admits the general expression

where η is the detection threshold and the generalized Marcum function is defined as

Some comments Given the detection structure, the signal at the input of the energy detector for RX i is

As a consequence the conditional (given the scattering) signal-to-interference ratio reads

Remark: the above represents the overall received SNR. There is an inherent trade-off between number of (transmitted) diversity dimensions and overall received signal-to-noise ratio, and rank-1 coding corresponds to concentrating all of the energy along one and the same direction.

Waveform design • This amounts to choosing the rank δ of the STC

matrix (or the number of active transmitters) and to deciding on how to distribute the energy budget among the transmit degrees of freedom.

• Question: does an optimal transmit policy exist? Answer: it depends on the figure of merit. For example, if SNR is the figure of merit, the optimum choice is obviously rank-1 coding along the least interfered direction, which means giving up angular diversity.

Figures of merit

• Figures involving the pair (PFA, PD); • The mutual information between the

observations available at the receive antennas and the scattering matrix of the target;

• Figures of merit involving Kullback-Leibler divergence;

• Figures of merit involving both the detection performance and the estimation accuracy, if some parameter must be estimated.

Detection and False-Alarm Probabilities

• Here it is necessary to resort to some hypotheses, in order to come up with closed-form results.

• Assumptions: Gaussian scattering; • Possible tools: maximize the Lower Chernoff

Bound (LCB) to the detection probability. Remark, however, that this bound is tight only in the large signal-to-noise ratio region, as detailed later.

LCB The LCB to the detection probability under Gaussian scattering reads

LCB-optimal codes

• Maximize the LCB to the detection probability subject to constraints on the rank of the STC matrix and the average received SNR (much questioned).

• Solution: generate as many independent and identically distributed diversity paths as possible.

Example

N=M=L=2 and white noise

Rank-1 coding and GLRT LCB-optimal and GLRT Rank-1 coding and energy detector

Comments

• Apparently, LCB-optimal codes are optimal only in the large SNR region, while rank-deficient coding is largely preferable in the low SNR region.

• This suggests (and theory confirms) that no uniformly optimum transmit policy exists if the figure of merit is detection performance.

Determining the transmit policy

Find the STC matrix maximizing the LCB subject to: a) A Semi-definite constraint on the matrix rank, i.e. Rank(C)=θ ≤

∆ b) Average received energy The above problem does not admit any solution which is optimal for any SNR, and evidence that an uniformly optimal policy does not exist if LCB is our major figure of merit.

Performance

Optimal rank of the code matrix (i.e., requiring minimum SNR to achieve the miss probability on the abscissas). Pfa=10-4; N=M=4;

Different conclusions

Mutual-Information (MI)-optimal codes (Gauss-Gauss and uncorrelated scattering)

Average received SNR:

Once again, θ denotes the rank of the STC matrix, and is a design parameter subject to a semi-definite constraint.

Different conclusions

• Subject to semi-definite constraints on the STC rank and the transmit (or receive, when applicable) energy

a) Maximize MI: Solution is diversity maximization, with a power allocation which attempts to create as many independent and identically distributed paths as possible

b) Maximize SNR: Here the solution is rank-1 coding along the least interfered direction.

Comments

• MI is a Schur-concave function of the variables it depends upon;

• Matrix trace is a Schur-convex function of the matrix eigenvalues;

• LCB is neither concave nor convex, but is not tight for low SNR, so it does not appear perfectly suited.

Memo: Schur-convexity/concavity

Any symmetric (i.e., invariant to argument permutation) function which is concave (convex) is also Schur-concave (Schur-convex). MI is Schur-concave, and attains its maximum when the arguments are maximally “dispersed”, i.e. when we generate as many independent and equivalent paths as possible through full-rank coding and water-filling. Trace is Schur-convex and is optimum for maximum “concentration” of its arguments, i.e. Rank-1 coding.

Divergence-based figures of merit Assume that we have to decide between two hypotheses, corresponding to two densities of the observations, say f1(x) and f0(x). We know that, the more “different” the densities, the easier to make correct decisions. A measure of such a difference is the pair

These two measures are relevant in testing f1 against f0, whether we resort to Fixed-Sample-Size Tests (FSST) or Sequential Probability Ratio Tests (SPRT).

FSST: Chernoff-Stein Lemma Let Xn be iid ∼ f0(x) or f1(x) and define PFA as the probability of declaring that f1(x) is in force as the alternative is true, while PM=1-PD is the probability of a miss. Then, among all the test achieving PFA≤α, the best that we can do has the following asymptotic property:

While D10 is the error exponent of the false alarm probability. Passing over the details, these two measures also have relevance in SPRT’s, regulating the Average Sample Number (ASN) needed to make decisions under the two hypotheses.

Sequential tests Here the number of observations is random and any performance can be obtained by collecting a large enough number of samples. The primary parameters are thus the average sample numbers (ASN’s) under the two alternatives. Sequential Probability Ratio Test (SPRT) aimed at discriminating between f0(x) and f1(x) enjoys the following:

Figures of merit

Divergence-based figures of merit have been proposed in Kay, IEEE Trans, AES, 2009, D01 adopted as foM Tang, Li, Wu, IEEE Signal Processing Letters, 2009, Grossi, Lops, IEEE Trans. Info. Theory, 2012, proposing the convex combination κD10+(1-κ)D01 In general, it is interesting to define, as a function of the waveforms that can be transmitted (i.e., in our setup, of the STC matrix) the achievability region, by constraining either the transmit energy or the received SNR. This gives us all of the information on what can be achieved, as far as detection is concerned, through STC in MIMO radar.

Achievability region, constrained received SNR

Achievability region, constrained received SNR

Achievability region, constrained transmit energy

An example of application: FSST

FSST, divergences chosen based on the achievability region.

An example of application: SPRT

Synopsis (∆=11)

The evidence speaks in favor of the fact that, in the low SNR region, we’d better use MIMO as an incoherent power multiplexer rather than as a source of diversity (or, equivalently, that we concentrate all of the power on one transmitter). κ=0 corresponds to assuming D01 as a measure of performance, and κ=1 to assuming D10.

Question Is there any realistic situation – concerning detection – wherein full angular diversity is desirable? Otherwise stated, if full diversity is advantageous only in the large SNR region, why to pay so much attention to it? A possible answer resides in the fact that, so far, we have operated under fairly ideal conditions, studying the fundamental limits with scarce attention to a such a basic issue as robustness, i.e. resilience of the detection performance with respect to deviations between the nominal and the actual operating conditions. These deviations may concern: The disturbance covariance; The amount of transmit diversity (we underline here that, for given wavelength, path independence is a function of the distance and of the target extension); The amount of receive diversity (same situation as above). It is thus interesting to focus attention on the issue of robust waveform design, which leads us to fairly different conclusions!!

Theoretical framework Let Mw = Qw⊗Rw be, as usual, the disturbance covariance and E[αiαi

H] =Qα⊗ Rα the clutter and scattering covariance, encapsulating the transmit diversity (matrices R) and the receive diversity (matrices Q, assumed IL for simplicity). The STC matrix C (or transmitted waveform) can be “robustified” under two different situations through the mini-max principle

Mini-max Waveforms All cost functions which are decreasing and Schur-convex admit one and the same robust solution, i.e. full transmit diversity. The power allocation depends on the operating conditions, i.e.: Water-filling, if M is known, assigning more power to the least interfered directions; Isotropic transmission, if M is unknown.

Interpretation: No empty sub-space exists where the target may hide. What matters here is the transmit policy (i.e., the amount of transmit diversity), while the power allocation strategy is definitely less critical. Problem: Do cost functions fulfilling the above conditions have any relevance in radar detection/estimation?

Admissible cost functions Are in the required form the following families of cost functions Any decreasing function of the received SCR; The LMMSE estimator of the target scattering matrix; The mutual Information between the observations and the target scattering. Moreover, since:

also the probability of miss will do, at least in the two relevant regions of large and small SCR.

Remark

The previous derivations assumed full receive diversity and partially or totally unknown transmit diversity. Everything carries over to the general case where both scattering and clutter have unknown transmit and receive diversity, and the major results re-apply to the cases where the overall covariance matrices take on the form of a Kronecker product between transmit and receive covariance matrices.

Example of application: effect of transmit diversity

M=L=2 (Number of receive and transmit antennas), N=4 (code-word length) Rα=Uα Λα Vα

H, Λα =σ2αdiag(ρ α ,1-ρ α)

Rw=Uw Λw Vw

H

Legenda for next slides: Optimal Coding: everything is known; Mismatched coding: the receiver assumes ρ α =0, and the signal is built around the absence of transmit diversity.

The typical figures of merit

More figures of merit

Major role of (angle) diversity in detection

At the transmitter end: full angle diversity, which may be achieved through full-rank coding for un-resolvable paths or by distributing the available power among a number of wide-band transmitters, for the case of resolvable paths, is fundamental in granting robustness in almost all of the most credited figures of merit. At the receiver end, of course diversity is always beneficial, which is a well-known results concerning decentralized systems. The above holds true for detection (and, somehow, parameter estimation if LMMSE is considered). What about other issues like localization? This is a long story, and we do not have much time, but we can give some general trends.

What about localization? • Two main situations:

– Un-resolvable paths: target ranging – Resolvable paths: target localization through multi-

lateration. The latter has been thoroughly studied in Godrich, Haimovich and Blum, “Target Localization accuracy

gain in MIMO radar-based systems”, IEEE Trans. Info. Theory, 2010.

Godrich, Petropolu, Poor, “Power Allocation Strategies for Target Localization in Distributed Multiple-Radar Architectures”, IEEE Trans. Signal Processing, 2011.

Target Ranging in un-resolvable paths

Assume a space-time coded MIMO system, transmitting the waveforms

Fisher information in delay estimation

This is a Schur-convex function, which is maximum for rank-1 coding.

Consequence If f (Σ) is any figure of merit, no matter its Schur-concavity/convexity characteristic, an optimization problem in the form

Cannot but imply rank-deficient coding. In other words, angle diversity maximization is not optimal except that if the figure of merit is Schur-concave and the constraint marginal (i.e., the value of β is small).

Resolvable paths (M×L MIMO) Localization in a plane of a target located at (x,y); Deterministic channels.

For details, see Godrich, Haimovich, Blum, IEEE Transactions on Information Theory, 2010.

Facts: othe (measurable) delays τi,j are known functions of (x,y), once the TX and RX positions are known. oAny vector parameter θ containing (x,y) as its first two entries has FIM J(θ). oThe 2×2 north-western sub-matrix of J-1(θ), say D(θ), has, on its diagonal elements, the CRLB to the estimates of x and y. oThe key cost function is thus Trace(D).

Design parameters and constraints

Cost function

Some allocation criteria

Example

•5 transmitters and 5 receivers •Available bandwidth: 3 MHz •PRT= 5 kHz •4 stationary equal-strength targets with RCS=10m2; •The targets are randomly located in a 20km×20km area •The channels are Rayleigh and random. •1000 trials

The cost function

Achieved performance

Resolvable paths: amount of transmit diversity

Resolvable paths: effect of the optimization

Some comments

The previous plots refers to un-equal average SNR, each channel being characterized by a Rayleigh attenuation with unit mean square value;

Once again angular diversity appears more a means for robustifying localization procedure than a tool to improve performance, as clearly demonstrated by the fact that the “best” (i.e., most frequently chosen) transmit policy does not rely on transmit diversity maximization.

A possible conclusion is that, for both detection and localization, the amount of transmit diversity must be a compromise between optimality and robustness, on the understanding that in many application maximizing the transmit diversity is a mini-max (or maxi-min, if a merit function is involved) choice.

Some references: angular diversity models and diversity

1. E. Fishler, A. Haimovich, R. Blum, L. Cimini, D. Chizhik, and R.

Valenzuela, “Spatial diversity in radars—Models and detection performance,” IEEE Trans. Signal Processing, vol. 54, pp. 823–838, Mar. 2006.

2. A. Haimovich, R. Blum, and L. Cimini, “MIMO radar with widely separated antennas,” IEEE Signal Processing Mag., vol. 25, pp. 116–129, Jan. 2008.

3. F. Daum and J. Huang, “MIMO radar: Snake oil or good idea,” IEEE Aerosp. Electron. Syst. Magazine, pp. 8–12, May 2009.

Remark: the last is kind of critical with respect to angular diversity, but offers an interesting point of view.

References on Optimal Waveform design

1. Y. Yang and R. S. Blum, ”MIMO Radar Waveform Design Based on Mutual Information and Minimum Mean-Square Error Estimation”, IEEE Transactions on Aerospace and Electronic Systems, Vol. 43, No. 1, January 2007, pp. 330-343.

2. A. Maio and M. Lops, “Design principles of MIMO radar detectors,” IEEE Trans. Aerosp. Electron. Syst., vol. 43, pp. 886–898, July 2007.

References on trade-offs and diversity (detection)

1. E. Fishler, A. Haimovich, R. Blum, L. Cimini, D. Chizhik, and R. Valenzuela, “Spatial diversity in radars—Models and detection performance,” IEEE Trans. Signal Processing, vol. 54, pp. 823–838, Mar. 2006.

2. A. De Maio, M. Lops, L. Venturino, “Diversity-integration trade-offs in MIMO detection,” IEEE Transactions on Signal Processing, vol. 56, no. 10, pp. 5051-5061, October 2008.

3. Q.He and R. Blum, “Diversity Gain for MIMO Neyman-Pearson Detection”, IEEE Transactions on Signal Processing, vol. 59, no. 3, pp. 869-881, March 2011.

References on different figures of merit

1. S. Kay, “Waveform design for multistatic radar detection,” IEEE Trans. Aerosp. Electron. Syst., vol. 45, no. 3, pp. 1153–1166, Jul. 2009.

2. J. Tang, N. Li, Y. Wu, and Y. Peng, “On detection performance of MIMO radar: a relative entropy-based study,” IEEE Signal Processing Lett., vol. 16, no. 3, pp. 184–187, Mar. 2009.

3. E. Grossi and M. Lops, "Space-time code design for MIMO detection based on Kullback-Leibler divergence," IEEE Trans. Inf. Theory, vol. 58, no. 6, pp. 3989-4004, Jun. 2012.

References on Robust Design

1. Y. Yang and R. S. Blum, ”Minimax Robust MIMO Radar Waveform Design”, IEEE Journal on Selected Topics on Signal Processing, Vol. 1, No. 1, June 2007, pp. 147-155.

2. E. Grossi, M. Lops, L. Venturino, “Robust waveform design for MIMO radars,” IEEE Transactions on Signal Processing, July 2011.

3. E. Grossi, M. Lops, and L. Venturino, "Min–max waveform design for MIMO radars under unknown correlation of the target scattering," Signal Process., vol. 92, no. 6, pp. 1550-1558, Jun. 2012.

MIMO Localization/Ranging

1. H. Godrich, A. Haimovich, R.S. Blum, ”Target Localization Accuracy Gain in MIMO Radar-Based Systems”, IEEE Transactions on Information Theory, Vol. 56, No.6, June 2010.

2. H. Godrich, A. Petropulu, and H. V. Poor, "Power allocation strategies for target localization in distributed multiple radars architectures", IEEE Transaction on Signal Processing, vol. 59, no. 7, July 2011.

3. N. Garcia, A. Haimovich, M. Coulon, M. Lops, “Resource Allocation in MIMO Radar With Multiple Targets for Non-Coherent Localization”, IEEE Transaction on Signal Processing, vol. 62, no. 10, May 2014, pp. 2656 - 2666.

4. G. H. Jajamovich, M. Lops, X. Wang, “Space-Time Coding for MIMO Radar Detection and Ranging”, IEEE Transaction on Signal Processing, vol. 58, no. 12, Dec. 2010, pp. 6195-6206.

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